I'm not going to lie to you, there are a lot of materials you are going to want to buy to complete this guide.
Since patch 2.4 was implemented you can now train up to Artisan Alchemy at any trainer available. To get your skill past 300 you will need to see the Master Alchemy Trainer that is located in each factions appropriate starting city for the Burning Crusade in Hellfire Peninsula, Honor Hold and Thrallmar.
This is the easiest step for leveling your alchemy. This will get your alchemy to 60 and you'll end up with a lot of minor healing potions to give alts or use incase of an emergency. Psyche!! Keep these potions because they will be used to make lesser healing potions.
Oh joy!! More healing potions, I hope you have a lot of alts or low level friends. You'll be making Lesser Healing Potions for 50 levels and then Healing Potions for 30 more levels. Make sure you stock up on Briarthorn because that is the main component of each recipe.
This is the first part of the guide where you will start making less of each item while you're increasing your skills. From 140-210 you're going to be making three different items. Ensure you buy enough Stranglekelp, Mage Royal, Liferoot, and Gold thorn. You will get to make something other than just healing potions in this step…mana potions!!! YAY!! You'll also get to make elixir of agility, these can sell decently on the auction house.
Artisan alchemy will require you to make 6 different items to go from 210-300. This is where it can get expensive and sometimes intense. This step is oftentimes more expensive then going from 300-360 because the supplies are less available due to the lack of pre 60's farming them. Sungrass is your friend and will be used in 3 different parts. I recommend checking the auction house for these supplies while you are leveling up from 1-210.
Now that your 300, it's time to max your Alchemy to 375 and get your money out of it. The first three items you will be making require a lot of Felweed, so stock up in anyway you can. Alchemy is one of the easier crafting professions to level to 375, others required hard to get reagents for the most readily available recipes, but the availability was increased through patch 2.4.
The Major Dreamless Sleep Motion can be purchased at Allerian Stronghold from Leeli Longhaggle if you are alliance or at Thunderlord Stronghold from Daga Ramba if you are Horde.